We first note, by Theorem REF  and assumption (iii), that we have \(\operatorname{Supp}_R(\widehat{\operatorname{Tor}}_i^R(M,N))\subseteq \operatorname{\mathcal {Z}}\)  for all \(i\in \mathbb {Z}\) . Hence, by Theorem REF  and assumption (ii), we get \(\widehat{\operatorname{Tor}}_{-i}^R(M,N)=0\)  for all \(n-c+1\le i\le n\) .
Consider to the following exact sequence \(0\rightarrow L\rightarrow X\rightarrow M\rightarrow 0\)  where \(X\)  is totally reflexive and \(\operatorname{pd}_R(L)<\infty \) . By Theorem REF (i) we have \(\widehat{\operatorname{Tor}}_i^R(L,N)=0\)  for all \(i\in \mathbb {Z}\) . Therefore, the above exact sequence induces the following isomorphism \(\widehat{\operatorname{Tor}}_i^R(M,N)\cong \widehat{\operatorname{Tor}}_i^R(X,N)\)  for all \(i\in \mathbb {Z}\)  (see for example [1]}). Set \(Y:=\Omega ^{-(n+1)}X\) . Note that \(Y\)  is totally reflexive and \(X\approx \Omega ^{n+1}Y\) . In view of Theorem REF  we obtain the following isomorphisms:
\(\operatorname{Tor}_i^R(Y,N)\cong \widehat{\operatorname{Tor}}_i^R(Y,N)\cong \widehat{\operatorname{Tor}}_{i-n-1}^R(X,N)\cong \widehat{\operatorname{Tor}}_{i-n-1}^R(M,N)=0 \text{ for all } 1\le i\le c.\qquad \mathrm {(\ref {t11}.1)}\) 
It follows from (REF .1) and [2]} that \(\widehat{\operatorname{Tor}}_i^R(Y,N)=0\)  for all \(i\in \mathbb {Z}\) . Equivalently, it follows that \(\widehat{\operatorname{Tor}}_i^R(M,N)\cong \widehat{\operatorname{Tor}}_i^R(X,N)=0\)  for all \(i\in \mathbb {Z}\) . Thus, by Theorem REF (i) and assumption (ii), we see that \(\operatorname{H}^i_{\operatorname{\mathcal {Z}}}(M\otimes _RN)=0\)  for all \(0\le i\le n\) . In other words, \(\operatorname{\textnormal {grade}}_R(\operatorname{\mathcal {Z}},M\otimes _RN)>n\) . Also, by Theorem REF (iv) we have \(\operatorname{Tor}_{i}^R(M,N)\cong \widehat{\operatorname{Tor}}_i^R(M,N)=0\)  for all \(i\gg 0\) . Now the last assertion follows from assumptions (i), (iii) and the dependency formula (REF .1).
